The remaining service time upon reaching a high level in M/G/1 queues

The distribution of the remaining service time upon reaching some target le vel in an M/G/1 queue is of theoretical as well as practical interest, In g eneral, this distribution depends on the initial level as well as on the ta rget level. say, B. Two initial levels are of particular interest, namely, level "1" (i.e., upon arrival to an empty system) and level "B - 1" (i.e., upon departure at the target level). In this paper. we consider a busy cycle and show that the remaining service time distribution. upon reaching a high level B due to an arrival, converg es to a limiting distribution for B --> infinity. We determine this asympto tic distribution upon the "first hit" (i.e., starting with an arrival to an empty system) and upon "subsequent hits" (i.e., starting with a departure at the target) into a high target level B. The form of the limiting (asympt otic) distribution of the remaining service time depends on whether the sys tem is stable or not. The asymptotic analysis in this paper also enables us to obtain good analytical approximations of interesting quantities associa ted with rare events, such as overflow probabilities.